Schedule for: 19w5142 - Soft Packings, Nested Clusters, and Condensed Matter

In materials science, auxetic behavior refers to the rather unusual property of a material to laterally expand, rather than shrink, when stretched in some direction. Only sporadic examples are known. I will present a brief survey of the geometric theory of auxetic behavior for periodic bar-and-joint frameworks, as introduced in a series of recent papers by Ciprian Borcea and myself. The theory leads to new principles for designing an abundance of frameworks that provably support auxetic deformations. It also leads to effective algorithms for deciding when a given framework allows, infinitesimally, an auxetic deformation. We have developed code to effectively test for the auxetic property on frameworks built from crystal databases. I will conclude with a summary of the current challenges and some preliminary results. This is joint work with Ciprian Borcea (Rider U.) and Juan Castillo (Harvard U.). Our project was supported by a 2018-19 Harvard Radcliffe IAS Fellowship.

Rigidity Theory is concerned with the rigidity and flexibility analysis of bar-joint frameworks and related constraint systems of geometric objects. This area has a rich history which can be traced back to classical work of Euler, Cauchy and Maxwell on the rigidity of polyhedra and skeletal frames. In the first part of this talk I will give an introduction to the theory of local and global rigidity of bar-joint frameworks and related structures such as body-bar and body-hinge frameworks, and provide an overview of the key results. In the second part of the talk I will discuss the recent progress on extending these results to infinite periodic frameworks. I will also discuss some key open questions in this area, some of which have potential applications in crystallography, materials science and condensed matter physics.

Authors: Sangmin Lee, Erin G. Teich, Michael Engel, and Sharon C. Glotzer Complex crystallization pathways occur in a variety of systems both in nature and in simulations and experiments. These systems transition from the fluid phase to the solid phase, not via classical nucleation and growth, but rather through the emergence of single or multiple structural precursors in the fluid, which then give rise to the crystallization of the solid phase. The influence of these precursors on the solid phase crystallization, and the structural characteristics of the prenucleation phases, have yet to be fully elucidated. Here, we report three instances of two-step crystallization of hard-particle fluids, in which crystallization proceeds via a high-density precursor fluid phase with prenucleation motifs in the form of clusters, fibers and layers, and networks, respectively. These are motifs of varying dimension and complexity. We explore the influence that the dimension of these prenucleation motifs has on the crystallization process, and we structurally and dynamically characterize each crystallization event. The crystals that form are complex, including, notably, a crystal with 432 particles in its cubic unit cell. Our results establish the existence of two-step crystallization pathways in entropic systems and showcase the accompanying variety of prenucleation structures that are possible.

A {\it soft ball packing of unit balls with outer radius $\lambda > 0$}, is a family of pairs of concentric closed balls, one with radius $1+\lambda$, called a {\it soft ball}, and the other one with unit radius, called the {\it core} of the soft ball. In this talk I review results related to soft ball packings of unit balls, and some related concepts. Most of the presented results are joint work with K. Bezdek.

In this talk I am going to discuss a well-known connection between lattices in $\mathbb{R}^d$ and convex polytopes, parallelohedra, that tile $\mathbdd{R}^d$ with translations only. My main topic will be the Voronoi conjecture, a century old conjecture which is, while stated in very simple terms, still open in general. I plan to survey certain known results on the Voronoi conjecture and give a quick insight on a recent proof of the Voronoi conjecture in five-dimensional case. The talks is based on joint works with M.~Dutour-Sikiri\'c, A.~Gavrilyuk, A.~Magazinov, A.~Sch\"urmann, and C.~Waldmann.

Local detection of a global property in a geometric or combinatorial structure is usually a challenging problem. The classical Local Theorem for Tilings says that a tiling of Euclidean d-space is tile-transitive (isohedral) if and only if the large enough neighborhoods of tiles (coronas) satisfy certain conditions. This is closely related to the Local Theorem for Delone Sets, which locally characterizes those sets among uniformly discrete sets in d-space which are orbits under a crystallographic group. Both results are of great interest in crystallography. We discuss old and new results from the local theory of tilings and Delone sets.

We study combinatorial properties of families of simple 3-dimensional polytopes defined by their cyclic and strongly cyclic k-edge-connectivity. Among them are flag polytopes and Pogorelov polytopes, which are polytopes realizable in the Lobachevsky (hyperbolic) space L3 as bounded polytopes of finite volume with right dihedral angles. The latter class contains fullerenes — simple 3-dimensional polytopes with only pentagonal and hexagonal faces. We focus on combinatorial constructions of families of polytopes from a small set of initial polytopes by a given set of operations. Here we will present the classical result by V.Eberhard (1891) for all simple 3-polytopes, more recent results by A.Kotzig (1969), D.Barnette (1974, 1977), J.Butler (1974), T.Inoue (2008), and V.D.Volodin (2011), and their improvements by V.M.Buchstaber and the author (2017-2019). For fullerenes we have a more strong result. We also study polytopes realizable in L3 as polytopes of finite volume with right dihedral angles. On the base of E.M. Andreev's theorem (1970) we prove that cutting off ideal vertices defines a one-to-one correspondence with strongly cyclically 4-edge-connected polytopes different from the cube and the pentagonal prism. We show that any polytope of the latter family is obtained by cutting off a matching of a polytope from the same family or the cube with at most two nonadjacent orthogonal edges cut producing all the quadrangles. We refine D.Barnette's construction of this family of polytopes and give its application to right-angled polytopes. We refine the construction of ideal right-angled polytopes by edge-twists described in the survey by A.Yu.Vesnin (2017) on the base of results by I.Rivin (1996) and G.Brinkmann, S.Greenberg, C.Greenhill, B.D.McKay , R.Thomas, and P.G.Wollan (2005), and analyse its connection to D.Barnette's construction via perfect matchings. We make a conjecture on behaviour of volume under operations generalizing results by T.Inoue (2008) and give arguments confirming it.

As early as 1985, Landau free-energy models [1-3] and density-functional mean-field theories [4] were introduced in an attempt to explain the stability of quasicrystals, with only partial success if any. It is only in recent years, that great progress has been made in understanding the thermodynamic stability of quasicrystals in such simple isotropic classical field theories. Much of this has happened thanks to insight from the experimental observation of quasicrystalline order in diverse systems ranging from fluid dynamics to soft condensed matter. The key to unlocking the stability puzzle was in the realization that more than a single length scale was required, but more importantly in figuring out how to introduce these multiple scales into the models, and identifying the remaining requirements [5,6]. We and others have since managed to produce Landau and other mean-field theories with a wide range of quasicrystals as their minimum free-energy states, and have also confirmed some of these theories using molecular dynamics simulations with appropriately designed inter particle potentials [7-12]. I shall give a quick overview of the quasicrystals that can be stabilized in these theories—in systems of one or two types of particles, in two and in three dimensions—and attempt to identify a trend that might be emerging in going from Landau theories to more realistic density-functional mean-field theories. It remains an open question whether this trend may eventually lead to understanding the stability of quasicrystals in complex metallic alloys. This research is supported by Grant No. 1667/16 from the Israel Science Foundation. [1] P. Bak, Phys. Rev. Lett. 54, 1517 (1985). [2] N.D. Mermin, S.M. Troian, Phys. Rev. Lett. 54, 1524 [Erratum on p. 2170] (1985). [3] P.A. Kalugin, A.Yu. Kitaev, L.C. Levitov, JETP Lett. 41, 145 (1985). [4] S. Sachdev, D.R. Nelson, Phys. Rev. B 32, 4592 (1985). [5] R. Lifshitz, H. Diamant. Phil. Mag. 87, 3021 (2007). [6] R. Lifshitz, D. Petrich, Phys. Rev. Lett. 79, 1261 (1997). [7] K. Barkan, H. Diamant, R. Lifshitz, Phys. Rev. B 83, 172201 (2011). [8] A.J. Archer, A.M. Rucklidge, E. Knobloch, Phys. Rev. Lett. 111, 165501 (2013). [9] K. Barkan, M. Engel, R. Lifshitz, Phys. Rev. Lett. 113, 098304 (2014). [10] C. V. Achim, M. Schmiedeberg, and H. Löwen, Phys. Rev. Lett. 112, 255501 (2014). [11] P. Subramanian, A.J. Archer, E. Knobloch, A.M. Rucklidge, Phys. Rev. Lett. 117, 075501 (2016). [12] S. Savitz, M. Babadi, R. Lifshitz, IUCrJ 5, 247 (2018). [13] M.C. Walters, P. Subramanian, A.J. Archer, R. Evans, Phys. Rev. E 98, 012606 (2018). [14] S. Savitz, R. Lifshitz, “Self-assembly of body-centered icosahedral cluster crystals”, presentation at this conference (2019).

Authors: Jaeuk Kim and Salvatore Torquato\\ Disordered hyperuniform packings (or dispersions) are unusual amorphous states of two-phase materials that are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in disordered hyperuniform materials, a major obstacle has been an inability to produce large samples that are perfectly hyperuniform due to practical limitations of conventional numerical and experimental methods. To overcome these limitations, we introduce a general theoretical methodology to construct perfectly hyperuniform packings in d-dimensional Euclidean space. Specifically, beginning with an initial general tessellation of space by disjoint cells that meets a “bounded-cell” condition, hard particles are placed inside each cell such that the volume fraction of this cell occupied with these particles becomes identical to the global packing fraction. We prove that the constructed packings with a polydispersity in size are perfectly hyperuniform in the infinite-sample-size limit. We numerically implement this procedure to two distinct types of initial tessellations; Voronoi and sphere tessellations. Beginning with Voronoi tessellations, we show that our algorithm can remarkably convert extremely large nonhyperuniform packings into hyperuniform ones in two and three dimensions. Application to sphere tessellations establishes the hyperuniformity of the classical Hashin-Shtrikman coated-spheres structures that possess optimal effective transport and elastic properties.

Building quasicrystal models with Zome tools, we discovered surprising connections between soft-packed clusters and Federov's famous parallelohedra. Parallelotopes, by definition, tile ${R}^n$ by translation and, therefore, their symmetries can only be those of lattices. But if we drop the (sometimes implicit) requirement of convexity, we find that parallelohedra in ${R}^3$ and clusters with "forbidden" icosahedral symmetries interlock. This raises perplexing questions about the role of convexity in the classical parallelotope theory (which I will briefly review): what, in fact, is that role? And how might we characterize parallelotopes if we drop it? (There's more than one answer to that.) These connections may also shed light on quasicrystal formation, as Jean Taylor will explain. (This is joint work with Jean and others in our SQuaRE at AIM.)

Jean Taylor, currently at University of California, Berkeley. Marjorie and I are interested in how crystals grow, especially how quasicrystals grow. In the case of multi-element alloys, it is highly likely to be by formation and then aggregation of clusters. There are many reasons to suspect icosahedral order may be important in forming many clusters; periodicity or quasiperiodicity would then arise from how these local clusters aggregate. Overlapping rhombic triacontahedra (RTs) are central to describing physical crystals with local or global icosahedral symmetry; structures derived from these overlapping RTs are a key to understanding how such crystals might grow and to unifying their presentations. Might overlapping rhombic icosahedra (RIs) also be useful to understanding how crystals and quasicrystals with local or global 5-fold symmetry can be described? Perhaps they could also be a key to how they might grow? I will describe both our suggestions as to how this might all work out and our search for evidence in physical crystals and quasicrystals.

Zeolites (zeo = boil; lithos = stone) are microporous, aluminosilicate minerals commonly used as commercial adsorbents and catalysts. In 1973 the book "Zeolite molecular sieves: structure, chemistry and use" (771pp) by Donald W. Breck appeared. At that time 27 zeolite framework types were known. In 2007, the 6'th edition of the Atlas of Zeolite Framework Types describes 176 known distinct approved types. Zeolites occur naturally but are also produced industrially on a large scale. Combinatorially, zeolites are line graphs of 4-regular graphs and may be finite. However, it is not always possible to realize a combinatorial zeolite as a unit distance graph in 3-space. Mike Winkler found a saturated tetrahedral packing consisting of only 12 tetrahedra. It is not known yet if the packing is rigid. A 16 vertex model was shown to have at least two degrees of freedom. We try to give an overview of recent results and open problems.

We enumerate and classify all stationary logarithmic configurations of d+2 points on the unit (d-1)-sphere in d-dimensions. In particular, we show that the logarithmic energy attains its relative minima at configurations that consist of two orthogonal to each other regular simplices of cardinality m and n, where m+n=d+2. The global minimum occurs when m=n if d is even and m=n+1 otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on the (d-1)-sphere for all d. The other two classes known in the literature, the regular simplex and the cross polytope, are both universally optimal configurations. This is a joint work with Peter Dragnev.

During the last three decades several results where obtained on bounded displacement equivalence of Delone sets. Two Delone sets are called bounded displacement equivalent if there is a bijection between them such that the distance between any two pairs is bounded by a common constant. (Or more generally: up to scaling) The case of crystallographic Delone sets was settled already in 1991: up to saling, all crystallographic Delone sets are in the same equivalence class. Hence interesting questions arise about the equivalence of nonperiodic Delone sets. A cute method to produce a large class of nonperiodic Delone sets are tile substitutions. In this talk we will explain the concepts above in detail and present several results on (non)-equivalence of Delone sets arising from substitution tilings.

Given a sufficiently nice embedded space curve, we can thicken it into a physical rope. A pair of physical configurations is Gordian if there is no physical isotopy that takes one to the other. It is an (old) open problem to describe a Gordian unknot. We will explore some special configurations of physical links by considering extrusion along with various packing constraints. This is a dual perspective to the topological sweep-out procedure Coward and Hass used to first describe a Gordian split link. There are some advantages that appear with our shifted view; we trade a general statement about knots and surfaces for tighter area bounds and rigidity, severely constraining the character of certain physical isotopies. In the end, we claim this is sufficient to describe a Gordian Unlink. Related to work with Rob Kusner, Greg Buck.

One of the methods in the design of new materials is to predict the crystal structure of a new compound from it’s molecular composition. This process involves generating many hypothetical structures based on lattice energy optimization. A different approach based only on the geometry of a molecule with it’s potential to speed up classical crystal structure prediction computations will be presented. Our preliminary results with regard to the periodic packing of a geometric representations of the pentacene molecule using Monte-Carlo molecular dynamic simulations will be also shown. Limitations and downsides of presented approach will be discussed, and future directions will be proposed.